Let Z be the set of all integers, A=(x,y) Z Z:(x-2)^2+y^24, B=(x,y) Z Z:x^2+y^24 and C=(x,…

Question

Let $\mathbb Z$ be the set of all integers, $A=\{(x,y)\in\mathbb Z	imes\mathbb Z:(x-2)^2+y^2\le4\}$, $B=\{(x,y)\in\mathbb Z	imes\mathbb Z:x^2+y^2\le4\}$ and $C=\{(x,y)\in\mathbb Z	imes\mathbb Z:(x-2)^2+(y-2)^2\le4\}$. If the total number of relations from $A\cap B$ to $A\cap B$ is $2^p$, then the value of $p$ is

Accepted Answer

$A\cap B$: integer points with $(x-2)^2+y^2\le4$ and $x^2+y^2\le4$.

- $x=0$: needs $y=0$ (1 point)
- $x=1$: $y\in\{-1,0,1\}$ (3 points)
- $x=2$: needs $y=0$ (1 point)
- $x=3,4$ (and negative $x$): no solutions

$$n(A\cap B)=1+3+1=5.$$

Relations from a $5$-element set to itself: $2^{5\times5}=2^{25}$, so $p=25$.

Solutions · 1